Mathematics

"At present game theory has, in my opinion, two important uses, neither of them related to games nor to conflict directly. First, game theory stimulates us to think about conflict in a novel way. Second, game theory leads to some genuine impasses, that is, to situations where its axiomatic base is shown to be insufficient for dealing even theoretically with certain types of conflict situations... Thus, the impact is made on our thinking process themselves, rather than on the actual content of our knowledge."

"Game theory, however, deals only with the way in which ultrasmart, all knowing people should behave in competitive situations, and has little to say to Mr. X as he confronts the morass of his problem."

"Game theory can be defined as the study of mathematical models of conflict and cooperation between intelligent rational decision-makers."

"Chapter 2 describes the most demanding rational choice theory of all, game theory, which was developed by a genius and assumes that other people are geniuses."

"Direct application of the theory of games to the solution of real problems has been rare, and its chief uses have been to offer some insight and understanding into the problems of competition (without actually solving them), and to provide mathematicians with new fields to conquer. Many important real problems involve more than two opponents, are not zero-sum, and exceed the bounds of the most developed versions of game theory."

"Game theory is logically demanding, but on a practical level, it requires surprisingly few mathematical techniques. Algebra, calculus, and basic probability theory suffice. ...the stress placed on game-theoretic rigor in recent years is misplaced. Theorists could worry more about the empirical relevance of their models and take less solace in mathematical elegance. ...[I]f a proposition is proved for a model with a finite number of agents, it is... irrelevant whether it is true for an ifinite number... There are... only a finite number of people, or even bacteria. Similarly, if something is true in games in which payoffs are finitely divisible... it does not matter whether it is true when payoffs are infinitely divisible. There are no payoffs in the universe... infinitely divisible. Even time... continuous in principle, can be measured only by devices with a finite number of s. Of course, models based on the real and complex numbers can be hugely useful, but they are just approximations... There is... no intrinsic value of a theorem that is true for a continuum of agents on a , if it is also true for a finite number of agents of a finite choice space."

"Game theory is about how people cooperate as much as how they compete... Game theory is about the emergence, transformation, diffusion and stabilization of forms of behavior."

"By the end of the war the new game theoretic methods that had been developed by von Neumann and Morgenstern were added to the toolkit and mathematical techniques that operations research scientists deployed. These proved very valuable, and game theoretic approaches took on great importance after the war."

"That strategic rivalry in a long-term relationship may differ from that of a one-shot game is by now quite a familiar idea. Repeated play allows players to respond to each other’s actions, and so each player must consider the reactions of his opponents in making his decision. The fear of retaliation may thus lead to outcomes that otherwise would not occur. The most dramatic expression of this phenomenon is the celebrated "Folk Theorem." An outcome that Pareto dominates the point is called individually rational. The Folk Theorem asserts that any individually rational outcome can arise as a in infinitely repeated games with sufficiently little discounting."

"Like all of mathematics, game theory is a tautology whose conclusions are true because they are contained in the premises."

"A proven theorem of game theory states that every game with complete information possesses a saddle point and therefore a solution."

"An equilibrium is not always an optimum; it might not even be good. This may be the most important discovery of game theory."

"Game theory is the study of individual decision-making in the face of competing boundedly rational actors."

"# General system theory in the narrower sense (G.S.T.), trying to derive from a general definition of “system” as complex of interacting components, concepts characteristic of organized wholes such as interaction, sum, mechanization, centralization, competition, finality, etc., and to apply them to concrete phenomena."

"# Factor analysis, i.e., isolation by way of mathematical analysis, of factors in multivariable phenomena in psychology and other fields"

"# or relational mathematics, including non-metrical fields such as network and graph theory."

"# , similarly analyzing rational choices, within human organizations, based upon examination of a given situation and its possible outcomes."

"# Game theory, analyzing in a novel mathematical framework, rational competition between two or more antagonists for maximum gain and minimum loss."

"# Information theory, introducing the concept of information as a quantity measurable by an expression isomorphic to negative entropy in physics, and developing the principles of its transmission."

"# Cybernetics, based upon the principle of feedback or circular causal trains providing mechanisms for goal-seeking and self-controlling behavior."

"There are quite a number of novel developments intended to meet the needs of a general theory of systems. We may enumerate them in brief survey:"

"Henry R. Towne is unquestionably the pioneer of management science. He began, as early as 1870, the systematic application at the Yale & Towne works, of what are now recognized as efficient management methods. In 1886, his paper "The Engineer as Economist," delivered before the American Society of Mechanical Engineers, probably inspired Frederick W. Taylor, then a young man of twenty, to devote his energies to the labor that formed his life work."

"The key to virtually every management science application is a mathematical model."

"The subject of management science has evolved for more than 60 years and is now a mature field within the broad category of applied mathematics. This book will emphasize both the applied and mathematical aspects of management science."

"The management science approach to organizational decision making is the analog to the rational approach by individual managers. Management science came into being during World War II. At that time, mathematical and statistical techniques were applied to urgent, large-scale military problems that were beyond the ability of individual decision makers. Mathematicians, physicists, and operations researchers used systems analysis to develop artillery trajectories, antisubmarine strategies, and bombing strategies such as salvoing (discharging multiple shells simultaneously). Consider the problem of a battleship trying to sink an enemy ship several miles away. The calculation for aiming the battleship's guns should consider distance, wind speed, shell size, speed and direction of both ships, pitch and roll of the firing ship, and curvature of the earth. Methods for performing such calculations using trial and error and intuition are not accurate, take far too long, and may never achieve success. This is where management science came in. Analysts were able to identify the relevant variables involved in aiming a ship's guns and could model them with the use of mathematical equations. Distance, speed, pitch, roll, shell size, and so on could be calculated and entered into the equations. The answer was immediate, and the guns could begin firing. Factors such as pitch and roll were soon measured mechanically and fed directly into the targeting mechanism. Today, the human element is completely removed from the targeting process. Radar picks up the target, and the entire sequence is computed automatically."

"Operations research has many precursors and allied fields, including Taylorism (after Frederick W. Taylor), scientific management and management science, industrial engineering and systems analysis. As one early textbook explained, the roots of OR "are as old as science and the management function. Its name dates back only to 1940" (Churchman et al. 1957: 3). Certainly its practitioners have expended much energy and ink in search of an acceptable definition of OR. Morse tried unsuccessfully to halt the debate by declaring OR to be 'the activity carried on by members of the Operations Research Society' (Morse 1953: 159) But his colleagues were not so easily dissuaded from debate. Much of the concern with definition focused on the sometimes elusive distinctions between OR and neighbouring fields; the attempt to define, or redefine, OR was also born of the desire to allow the subject to evolve beyond the orthodoxy of wartime experience. Crucial considerations included the balance between model and application, and the complexity of the mathematics involved."

"Now the salient characteristic of the decision tools employed in management science is that they have to be capable of actually making or recommending decisions, taking as their inputs the kinds of empirical data that are available in the real world, and performing only such computations as can reasonably be performed by existing desk calculators or, a little later electronic computers. For these domains, idealized models of optimizing entrepreneurs, equipped with complete certainty about the world - or, a worst, having full probability distributions for uncertain events - are of little use. Models have to be fashioned with an eye to practical computability, no matter how severe the approximations and simplifications that are thereby imposed on them... The first is to retain optimization, but to simplify sufficiently so that the optimum (in the simplified world!) is computable. The second is to construct satisficing models that provide good enough decisions with reasonable costs of computation. By giving up optimization, a richer set of properties of the real world can be retained in the models... Neither approach, in general, dominates the other, and both have continued to co-exist in the world of management science."

"During and after World War II, a large number of academic economists were exposed directly to business life, and had more or less extensive opportunities to observe how decisions were actually made in business organizations. Moreover, those who became active in the development of the new management science were faced with the necessity of developing decision-making procedures that could actually be applied in practical situations. Surely these trends would be conducive to moving the basic assumptions of economic rationality in the direction of greater realism."

"Decision theory can be pursued not only for the purposes of building foundations for political economy, or of understanding and explaining phenomena that are in themselves intrinsically interesting, but also for the purpose of offering direct advice to business and governmental decision makers. For reasons not clear to me, this territory was very sparsely settled prior to World War II. Such inhabitants as it had were mainly industrial engineers, students of public administration, and specialists in business functions, none of whom especially identified themselves with the economic sciences. Prominent pioneers included the mathematician, Charles Babbage, inventor of the digital computer, the engineer, Frederick Taylor and the administrator, Henri Fayol. During World War II, this territory, almost abandoned, was rediscovered by scientists, mathematicians, and statisticians concerned with military management and logistics, and was renamed “operations research” or “operations analysis.” So remote were the operations researchers from the social science community that economists wishing to enter the territory had to establish their own colony, which they called “management science”."

"When discussing management and science, I kept saying to myself over and over again that science could be looked at as a kind of management, or that management could be looked at as a kind of science. Saying that science can become a way of managing didn't imply automation or any other form of mechanical decision making, because none of this is science. Science is the creative discovery of knowledge. Management science is the process of trying to look at science as a management function. Similarly, management can be looked at as a scientific function, that is, as a way of finding out about the world."

"The aim of management science is to display the best course of action in a given set of circumstances, and this must include all the circumstances."

"The purpose of management science is to secure improvement in social systems by means of the scientific method."

"We have overwhelming evidence that available information plus analysis does not lead to knowledge. The management science team can properly analyse a situation and present recommendations to the manager, but no change occurs. The situation is so familiar to those of us who try to practice management science that I hardly need to describe the cases."

"In management science, it seems most likely to me that important developments will result from the efforts of research scientists working with management engineers and management people in their area. It is also possible that the accumulated, uncorrelated knowledge and experience of management people will have a valuable effort on any general general formulation concerning management.... Management science undoubtedly requires many disciplines, including mathematics, economics, psychology, sociology, engineering, and others. However, we believe that management science can also be defined as a separate science in its own right."

"There will be no drastic revolution in management functions or organizations in order to encompass systems management. Rather, the adaptation of systems management theory to organizations has been and will continue to be an evolutionary process."

"One cannot help but be struck by the diversity that characterizes efforts to study the management process. If it is true that psychologists like to study personality traits in terms of a person's reactions to objects and events, they could not choose a better stimulus than management science. Some feel it is a technique, some feel it is a branch of mathematics, or of mathematical economics, or of the "behavioral sciences," or of consultation services, or just so much nonsense. Some feel it is for management (vs. labor), some feel it ought to be for the good of mankind — or for the good of underpaid professors."

"The last decade has developed new and complex tools for management, in the form of high-speed computational, display, and communication techniques as well as of powerful analytical methods such as information theory, , and . These tools are being applied to continuing industrial problems, such as production and labor force scheduling, inventory control, sales forecast, and data processing (flow of information), as well as to special problems arising in connection with planning, market analysis, and organization. There is a rapidly growing demand for those well trained in the management sciences. Special graduate training in this held may be based on undergraduate study leading to a bachelor's degree in such fields as business administration, engineering, mathematics or the physical sciences, economics, and psychology."

"In the the past thirty years, what is now known as the science of has grown from a scattered, almost formless beginning into a well-defined, rapidly developing structure. Today this science bids fair to outstrip in public importance and economic usefulness any one of the great sciences that have been applied to serve mankind. For the science of industrial management is the great super-science that takes mechanics, chemistry, physics, electricity, psychology; and Burbank-like, grafts them all into one great, productive, ever-bearing plant."

"The Manager and the Instruments of Management Production is a synthesis of functions — always present but not all actually developed — sometimes quite rudimentary — how is synthesis to be effected — influence of strong personality — leadership essential to great results — manager is outside science of management — must use its principles, as chemist is outside the science of chemistry and uses its principles to effect his aims — management science provides the instruments, their successful wielding depends upon capacity of manager."

"[The decisionmaking role of the firm has progressed from the neoclassical standpoint of profit maximization to sales maximization, utility maximization, and satisficing. From the Operation Research point of view] ...the ideal picture is that someone, presumable the firm that hires the operations researcher, hands him, on a silver platter, an objective function. By talking to the engineers, or by looking into a few scientific laws, he determines the policy alternatives available and also the model"

"Operations research is a scientific approach to problem-solving for executive management."

"The development of information research has increased considerably the interaction of emerging information science with other disciplines. Librarianship has traditionally had links with education and classification and has drawn ideas from logic and philosophy. But during the last fifty years new insights and methods have been derived from sociology and social psychology, from computer science, from operations research and related quantitative approaches, from communications research, from linguistics, and most recently from the new hybrids: cognitive science and artificial intelligence."

"Herbert A. Simon's scientific output goes far beyond the disciplines in which he has held professorships: political science, administration, psychology and information sciences. He has made contributions in the fields of science theory, applied mathematics, statistics, operations research, economics and business and public administration (and), in all areas in which he has conducted research, Simon has had something of importance to say.""

"It may be said that systems engineering is directed at the design and operating problems of production processes and units, while operations research is applied to problems in other areas of management such as sales, marketing, and external finance."

"The conclusions of most good operations research studies are obvious."

"We should no longer have trouble explaining the scope and methods of operations research to the layman. We already can say: operations research is the activity carried on by members of the Operational Research Society; its methods are those reported in our journal."

"The Mathematical School ; Although mathematical methods can be used by any school of management theory, and have been, I have chosen to group under a school those theorists who see management as a system of mathematical models and processes. Perhaps the most widely known group I arbitrarily so lump are the operations researchers or operations analysts, who have sometimes anointed themselves with the rather pretentious name of "management scientists." The abiding belief of this group is that, if management, or organization, or planning, or decision making is a logical process, it can be expressed in terms of mathematical symbols and relationships. The central approach of this school is the model, for it is through these devices that the problem is expressed in its basic relationships and in terms of selected goals or objectives."

"It is hard to say whether increasing complexity is the cause or the effect of man's effort to cope with his expanding environment. In either case a central feature of the trend has been the development of large and very complex systems which tie together modern society. These systems include abstract or non-physical systems, such as government and the economic system. They also include large physical systems like pipe line and power distribution systems, transportation and electrical communication systems. The growth of these systems has increased the need not only for over-all planning, but also for long-range development of the systems. This need has induced increased interest in the methods by which efficient planning and design can be accomplished in complex situations where no one scientific discipline can account for all the factors. Two similar disciplines which emerged about the time of World War II to cope with these problems are called systems engineering and operations research."

"Operations research has many precursors and allied fields, including Taylorism (after Frederick W. Taylor), scientific management and management science, industrial engineering and systems analysis. As one early textbook explained, the roots of OR "are as old as science and the management function. Its name dates back only to 1940" (Churchman et al. 1957: 3). Certainly its practitioners have expended much energy and ink in search of an acceptable definition of OR. Morse tried unsuccessfully to halt the debate by declaring OR to be 'the activity carried on by members of the Operations Research Society' (Morse 1953: 159) But his colleagues were not so easily dissuaded from debate. Much of the concern with definition focused on the sometimes elusive distinctions between OR and neighbouring fields; the attempt to define, or redefine, OR was also born of the desire to allow the subject to evolve beyond the orthodoxy of wartime experience. Crucial considerations included the balance between model and application, and the complexity of the mathematics involved."

"By the end of the war the new game theoretic methods that had been developed by von Neumann and Morgenstern were added to the toolkit and mathematical techniques that operations research scientists deployed. These proved very valuable, and game theoretic approaches took on great importance after the war."